Optimal scale selection of dynamic incomplete generalized multi-scale fuzzy ordered decision systems based on rough fuzzy sets

Abstract

As a specialized form of decision systems, multi-scale decision systems play a crucial role in practical applications. The task of optimal scale selection for multi-scale decision systems is an important issue in the field of granular computing research, especially when data are dynamically updated. However, analyzing multi-scale fuzzy ordered information with dominance-based rough fuzzy sets poses significant challenges, particularly when handling incomplete data, which is prevalent in real-world scenarios. Meanwhile, multi-granulation rough fuzzy sets are an effective extension of rough set that describes a target concept from multiple perspectives. To address these challenges and explore multi-scale decision systems based on multi-granulation rough sets, this paper proposes a novel framework that integrates dominance-based rough fuzzy sets and multi-granulation rough sets into the analysis of dynamic incomplete multi-scale fuzzy ordered information. Firstly, we introduce an incomplete generalized multi-scale fuzzy ordered decision system, defining granular information transformation functions as monotonically increasing functions. By employing an expanded dominance relation, we analyze the changes in dominating and dominated classes across different scales and propose two types of optimal scale combinations (OSCs) along with an algorithm to identify them. Secondly, we extend the model by incorporating dominance-based multi-granulation rough fuzzy sets, and introduce four additional OSC concepts to enhance the analysis from a multi-granulation perspective. Furthermore, to handle dynamic data environments, we investigate the impact of adding new objects on six types of OSCs and design algorithms to update them efficiently. Numerical experiments demonstrate the effectiveness and practical applicability of the proposed framework, highlighting its potential to address complex decision-making problems involving incomplete and multi-scale fuzzy ordered information.